(
www.tiwariacademy.com
)XII
Free web support in Education 39
Exercise 3.3
Question 1:
Find the transpose of each of the following matrices:
(i)
Answer
(i)
(ii)
(iii)
( ii) ( iii)
(
www.tiwariacademy.com
)XII
Free web support in Education 40
Question 2:
If , then verify that
(i) (ii)
Answer We have:
(i)
(ii)
and
(
www.tiwariacademy.com
)XII
Free web support in Education 41
Question 3:
If , then verify that
(i)
(ii)
Answer
(i) It is known that Therefore, we have:
and
(
www.tiwariacademy.com
)XII
Free web support in Education 42
(ii)
(
www.tiwariacademy.com
)XII
Free web support in Education 43
Question 4:
If , then find
Answer
Question 5:
For the matrices A and B, verify that (AB)′ = where and
We know that
(
www.tiwariacademy.com
)XII
Free web support in Education 44
(i)
(ii)
Answer
(i)
(ii)
(
www.tiwariacademy.com
)XII
Free web support in Education 45
Question 6:
If (i) , then verify that
(ii) , then verify that Answer
(i)
(
www.tiwariacademy.com
)XII
Free web support in Education 46
(ii)
(
www.tiwariacademy.com
)XII
Free web support in Education 47
Question 7:
(i) Show that the matrix is a symmetric matrix
(ii) Show that the matrix is a skew symmetric matrix
Answer
(i) We have:
Hence, A is a symmetric matrix.
(ii) We have:
(
www.tiwariacademy.com
)XII
Free web support in Education 48
Hence, A is a skew-symmetric matrix.
Question 8:
, verify that
(i) (ii) Answer
( i )
Hence, is a symmetric matrix.
( ii )
Hence, is a skew-symmetric matrix.
For the matrix
is a symmetric matrix is a skew symmetric matrix
(
www.tiwariacademy.com
)XII
Free web support in Education 49
Question 9:
Find
Answer
and , when
(
www.tiwariacademy.com
)XII
Free web support in Education 50
Question 10:
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
The given matrix is
(
www.tiwariacademy.com
)XII
Free web support in Education 51
(i)
(ii)
(iii)
(iv)
Answer (i)
Thus, is a symmetric matrix.
(
www.tiwariacademy.com
)XII
Free web support in Education 52
Representing A as the sum of P and Q:
(ii)
Thus, is a skew-symmetric matrix.
(
www.tiwariacademy.com
)XII
Free web support in Education 53
Thus, is a symmetric matrix.
Thus, is a skew-symmetric matrix.
(
www.tiwariacademy.com
)XII
Free web support in Education 54
Representing A as the sum of P and Q:
(iii)
(
www.tiwariacademy.com
)XII
Free web support in Education 55
Thus, is a symmetric matrix.
Thus, is a skew-symmetric matrix.
(
www.tiwariacademy.com
)XII
Free web support in Education 56
Representing A as the sum of P and Q:
(iv)
Thus, is a symmetric matrix.
Thus, is a skew-symmetric matrix.
(
www.tiwariacademy.com
)XII
Free web support in Education 57
Representing A as the sum of P and Q:
Question 11:
If A, B are symmetric matrices of same order, then AB − BA is a A. Skew symmetric matrix B. Symmetric matrix
C. Zero matrix D. Identity matrix
Answer
The correct answer is A.
A and B are symmetric matrices, therefore, we have:
Thus, (AB − BA) is a skew-symmetric matrix.
Question 12:
If , then , if the value of α is
(
www.tiwariacademy.com
)XII
Free web support in Education 58
A.
B.
C. π D.
Answer
The correct answer is B.
Comparing the corresponding elements of the two matrices, we have:
